Homogenization of Elliptic Difference Operators

We develop some aspects of general homogenization theory for second order elliptic difference operators and consider several models of homogenization problems for random discrete elliptic operators with rapidly oscillating coefficients. More precisely, we study the asymptotic behavior of effective coefficients for a family of random difference schemes whose coefficients can be obtained by the discretization of random high-contrast checkerboard structures. Then we compare, for various discretization methods, the effective coefficients obtained with the homogenized coefficients for corresponding differential operators

Quantitative homogenization in a balanced random environment

We consider discrete non-divergence form difference operators in an i.i.d. random environment and the corresponding process--the random walk in a balanced random environment in $\mathbb{Z}^d$. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As a consequence, we quantify the quenched central limit theorem of the random walk with an algebraic rate. Furthermore, we prove algebraic rate of convergence for homogenization of the Dirichlet problems for both elliptic and parabolic non-divergence form difference operators.Comment: 36 pages, 1 figur